《误差理论与测量平差基础教学课件》第十讲.ppt
第二章第二章 参参 数数 平平 差差Parametric Least-Squares AdjustmentParametric Least-Squares Adjustment1、参数平差的函数模型、参数平差的函数模型误差方程误差方程复复 习习误差理论与测量平差基础误差理论与测量平差基础 测绘学院大地教研室 2006年 2、未知参数的估值方程、未知参数的估值方程法方程法方程3、参数平差的计算步骤、参数平差的计算步骤1 1)选参数；选参数；2)2)列出误差方程列出误差方程;3 3)组成法方程组成法方程;4 4）解算法方程解算法方程;5 5）计算改正数向量，观测量的平差值；计算改正数向量，观测量的平差值；6 6）精度估计。精度估计。当当P正定，正定，A列满秩时法方程有唯一解：列满秩时法方程有唯一解：1 1、Importing the parametric approximationImporting the parametric approximation2 2、Concept of Precision EstimateConcept of Precision Estimate3 3、Standard Error of Unit WeightStandard Error of Unit Weight第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate4 4、Calculate the VCalculate the VT TPVPV5 5、Weight Reciprocal and Standard Error of Weight Reciprocal and Standard Error of Functions of UnknownsFunctions of Unknowns1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate1 1）The necessaryThe necessary 参数一般是平差问题中的物理量；参数一般是平差问题中的物理量；如坐标，角度，距离等如坐标，角度，距离等这些量的数值一般都比较大；这些量的数值一般都比较大；有些量还是复名数，如角度；有些量还是复名数，如角度；计算起来麻烦，故引入近似值。计算起来麻烦，故引入近似值。1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate2 2）How to import the parametric approximationHow to import the parametric approximation 1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimateExample 1Example 1 以单三角形为例，设观测了三角形三内角，其观测值为以单三角形为例，设观测了三角形三内角，其观测值为626217175252，333352521919及及838349494343，求各角最或然值。，求各角最或然值。第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate1 Error equation1 Error equation2 Approximation2 Approximation3 Error equation3 Error equation1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate3 3）The consistent in import and no-importThe consistent in import and no-import不引入近似值不引入近似值引入近似值引入近似值1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate4 4）Linearing of no-linear equationLinearing of no-linear equation1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimateExample 2Example 21 Error equation1 Error equation2 Approximation2 Approximation3 Linear 3 Linear 1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate4 Coefficient4 Coefficient3 Error equation3 Error equation1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimateLinearization:Linearization:is the process of approximating is the process of approximating non-linear functions by a linear one.The non-linear functions by a linear one.The most convenient and efficient approach of most convenient and efficient approach of linearization used in Geomatics is Taylor linearization used in Geomatics is Taylor Series ExpansionSeries Expansion 1 1、Importing the parametric approximationImporting the parametric approximation第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimatePonder the following questions:Ponder the following questions:(1)(1)Can we select the approximation arbitrarily?Can we select the approximation arbitrarily?(2)(2)If we select adjustment value as the If we select adjustment value as the approximation,the and approximation,the and l l is?is?(3)(3)If we select true value of X as approximation,If we select true value of X as approximation,the and the and l l is?is?2 2、Concept of Precision EstimateConcept of Precision Estimate1)The necessary of precision estimate第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimatea a、one of the adjustment purposeone of the adjustment purposeb b、pay attention to distinguish between true pay attention to distinguish between true error and residual errorerror and residual error2)Content of precision estimate a a、standard error of unit weightstandard error of unit weightb b、weight reciprocal of function of parameterweight reciprocal of function of parameter3 3、Some weight inverse matricesSome weight inverse matrices1)Weight inverse matrix of l is equal to the observations.第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate3 3、Some weight inverse matricesSome weight inverse matrices第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate2)Weight inverse matrix of U is N3 3、Some weight inverse matricesSome weight inverse matrices第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate3)Weight inverse matrix of is the inverse matrix of N3 3、Some weight inverse matricesSome weight inverse matrices第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate4)Independence between V and adjustment value 改正数向量和所有平差值向量不相关改正数向量和所有平差值向量不相关3 3、Some weight inverse matricesSome weight inverse matrices5)Weight inverse matrix of adjustment value of observations第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate3 3、Some weight inverse matricesSome weight inverse matrices第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate6)Weight inverse matrix of V The precision of adjustment value is higher than The precision of adjustment value is higher than observationsobservations3 3、Some weight inverse matricesSome weight inverse matrices第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate平差值的精度高于观测值的精度；平差值的精度高于观测值的精度；在观测精度一定的情况下，在观测精度一定的情况下，v v的中误的中误差越大，平差值的中误差越小。差越大，平差值的中误差越小。第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate4 4、Calculate the VCalculate the VT TPVPV第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate4 4、Calculate the VCalculate the VT TPVPV公式的用途：公式的用途：（1 1）作为检核公式；）作为检核公式；（2 2）公式推导。）公式推导。5 5、Standard Error of Unit WeightStandard Error of Unit Weight第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate函数模型函数模型方法方法1 1得到得到于是于是5 5、Standard Error of Unit WeightStandard Error of Unit Weight第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate函数模型函数模型方法方法1 1顾及顾及于是于是5 5、Standard Error of Unit WeightStandard Error of Unit Weight第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate单位权方差单位权方差方法方法1 1单位权中误差单位权中误差5 5、Standard Error of Unit WeightStandard Error of Unit Weight第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate函数模型函数模型方法方法2 2得到得到于是于是5 5、Standard Error of Unit WeightStandard Error of Unit Weight第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate方法方法2 2两边取数学期望两边取数学期望于是于是5 5、Standard Error of Unit WeightStandard Error of Unit Weight第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate1 1、单位权中误差的平方是单位权方差的无偏估值、单位权中误差的平方是单位权方差的无偏估值NoteNote2 2、由残差、由残差V V计算计算u u称为验后单位权中误差（验后估称为验后单位权中误差（验后估计）；由真误差计）；由真误差计算计算u u称为验前中误差（验前估称为验前中误差（验前估计）。计）。3 3、平差计算的精度估计采用验后估计方法、平差计算的精度估计采用验后估计方法第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate6 6、Weight Reciprocal and Standard Error of Weight Reciprocal and Standard Error of Functions of UnknownsFunctions of Unknowns 测量中的实际问题：控制网平差中，以点的坐标测量中的实际问题：控制网平差中，以点的坐标为参数，平差后可得到参数的估值及其权逆阵，为参数，平差后可得到参数的估值及其权逆阵，实际应用中还需要知道方位角的精度，这就涉及实际应用中还需要知道方位角的精度，这就涉及到如何计算参数函数的权倒数和中误差的问题了到如何计算参数函数的权倒数和中误差的问题了第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate6 6、Weight Reciprocal and Standard Error of Weight Reciprocal and Standard Error of Functions of UnknownsFunctions of Unknowns第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimateExampleExampleAP2h3P1h1Bh2h4 如图所示水准网，如图所示水准网，A A、B B为已知点，为已知点，高程为高程为H HA A 、H HB B ，设为无误差，设为无误差，P1P1、P2P2为未知点为未知点 ,各观测的路线长分别为各观测的路线长分别为 S1=10kmS1=10km，S2=10km S2=10km，S3=20km S3=20km，S4=10km S4=10km。试求：（试求：（1 1）P1P1、P2 P2 点高程平差值的权倒数点高程平差值的权倒数;(2)(2)P1 P1 至至 P2 P2 点高差平差值的权倒数。点高差平差值的权倒数。第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimateExampleExampleAP2h3P1h1Bh2h42212AP2h3P1h1Bh2h42212法方程为法方程为于是于是，AP2h3P1h1Bh2h4于是于是，1、2间的高差间的高差作业：作业：2.2 2.3 2.5 2.6 2.12 2.13 2.14 2.16 2.18 2.2 2.3 2.5 2.6 2.12 2.13 2.14 2.16 2.18 2.20 2.26 2.27 2.28 2.292.20 2.26 2.27 2.28 2.29第十讲第十讲 参数近似值的引入及精度估计参数近似值的引入及精度估计Importing Parametric Approximation and Precision estimate
